well first of all that isn't true ...

you have the line ( I didn't check all your work up there) that says :

and you have there results and

if you are to put that solutions in that equation you get :

which isn't true

when done correctly you just put those A,B,C,D,E .... coefficient back to the equation and you will get partial fractions

note that you can always check your work, if you are just that what did you get (partial fractions) solve :

and you must get starting fraction ... if that's not the case than you made some errors while working